MATH 3160: APPLIED COMPLEX VARIABLES
FINAL EXAM (VERSION A)
Name:
This test has 7 pages and 6 problems.
No calculators are allowed. You are permitted only a pencil or pen.
Work out everything as far as you can before making decimal approximations.
1. Calculate
1
Arg
1+i
Date: May 2, 2001.
1
2 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A)
2. Is u(x, y) = e−y cos x the real part of an analytic function? If
so, find the imaginary part of that function.
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A) 3
3. (a) Give two nonzero terms in the Laurent expansion of
cos (1/z) sin (1/z)
about z = 0.
(b) What sort of singular point does this function have at z =
0?
(c) Find
Z
cos(1/z) sin(1/z)dz .
|z|=1
4 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A)
CR
Cρ
L1
L2
Figure 1. A contour, with ρ (the radius of Cρ ) small,
and R (the radius of CR ) large.
4. Show that
Z
∞
sin x
dx = π
−∞ x
by integration of the function
eiz
f (z) =
z
along a contour like that shown in figure 1.
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A) 5
5. Using the principal branch of the square root, let
√
Z = sin z .
Where does it map the half strip
0 < x < π/2, y > 0?
Draw the boundaries of this half strip, and show where they go
under this map.
6 MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A)
15
10
5
–1.5
–1
–0.5
0.5
x~
1
1.5
Figure 2. The stream lines of the flow whose complex
potential is F (z) = sin z
6. (a) Show that F (z) = sin z is the complex potential of a flow
that never leaves the half strip
−π/2 ≤ x ≤ π/2, y ≥ 0 .
(b) Calculate the equations for all of the stream lines. Write
them in the form of y as a function of x. (They should
look like the ones drawn in figure 2.)
(c) Show that the stream function is positive everywhere in
the half strip.
(d) Show from the definition of a stream line (level curves of
the stream function) that the only critical point of y as
a function of x on each stream line is at x = 0. Hint:
implicitly differentiate the equation ψ = c0 , where ψ is the
stream function.
(e) Show that the closest any stream line gets to the x axis is
at the point
q
2
x = 0, y = ln c0 + c0 + 1 .
MATH 3160: APPLIED COMPLEX VARIABLES FINAL EXAM (VERSION A) 7
(f ) Show that the stream line passing through the point
√
π
x = , y = ln 2
4
never goes closer to the x axis than a distance
√ !
1 + 17
ln
.
4